The Eighth Compositional Iterate Modulo Eight

Abstract

Every coefficient above degree one in the eighth compositional iterate of A396798 is divisible by eight.

Write I0(F)=X and I(k+1)(F)=Ik(F) composed with F. The source is an ordinary integer formal series satisfying A=X+I4(A)*I5(A), with zero constant coefficient. The product is ordinary series multiplication, and no factorial scaling is used. Let H(F)=X+I4(F)*I5(F). In the definition below, iterateFunction(H,k,0) means H applied k times to zero, and mk assembles a series from its coefficient function.

Definition 1.1 (The source series).

$$A=\mathrm{mk}\left(n\mapsto \mathrm{coeff}\left(n,\mathrm{iterateFunction}\left(H,n+1,0\right)\right)\right)$$

Formalization. D5/S1/Recurrence/Residue/IterateProductFourFiveEighthModEight.generatingSeries (✓ std3).

Source. Repository-derived.

Acknowledgement. Paul D. Hanna (2026). OEIS A396798: compositional iterates modulo eight. URL: https://oeis.org/A396798.

Commentary.

The n-th coefficient is taken from H iterated n+1 times at the zero series. The result’s proof establishes coefficient stabilization, the source equation and uniqueness among zero-constant fixed points.

Theorem 1.2 (Hanna’s eighth conjecture).

$$\forall n:\mathbb{N},1<n⟹8\mid \mathrm{coeff}\left(n,\mathrm{iterate}\left(A,8\right)\right)$$

Proof. Machine-checked in Lean as D5/S1/Recurrence/Residue/IterateProductFourFiveEighthModEight.result (✓ std3). ∎

Resolves. Problems/oeis-a396798-eighth-iterate-mod-eight (proved) by D5/S1/Recurrence/Residue/IterateProductFourFiveEighthModEight.result.

Source. Repository-derived.

Acknowledgement. Paul D. Hanna (2026). OEIS A396798: compositional iterates modulo eight. URL: https://oeis.org/A396798.

Commentary.

For every natural index n>1, the coefficient of the eighth compositional iterate is divisible by eight. Source uniqueness identifies A modulo four with X/(1-X). An exact integer quotient and square-zero iteration then yield the eighth iterate equal to X modulo eight. This resolves only the eighth comment of OEIS revision 12; the other comments retain their separate scopes and evidence.

References

• Truth anchor: D5/S1/Recurrence/Residue/IterateProductFourFiveEighthModEight.generatingSeries
• Truth anchor: D5/S1/Recurrence/Residue/IterateProductFourFiveEighthModEight.result
• Dependency: D5/S1/Recurrence/Invariants/ThreeFourIterateProductModSix
• Dependency: D5/S1/Recurrence/Parity/DiagonalIterateEven
• Dependency: D5/S1/Recurrence/Residue/IterateProductTwentyFiveModFive